Consider two continuous random variables $X,Y$ with joint, conditional, and marginal densities (respectively) $f(x,y), f(y|x), f(x)$. Let $A,B$ be two intervals.
Do we have the following?
$$Pr(Y\in A|X\in B)=\int_A \int_B f(y|x) dxdy$$
I know that by Bayes's rule, we have $Pr(Y\in A|X\in B)=\frac{\int_A \int_B f(y,x) dxdy}{\int_B f(x) dx}$. But I am not sure whether this can be written sololy as the integral of the conditional density function.
Any counter-examples is welcomed.
If $X$ and $Y$ are independent, then $f(y \mid x) = f(y)$, so the right-hand side is $|B| \cdot P(Y \in A)$ while the left-hand side is $P(Y \in A)$.